Orthogonal Polynomials in Information Theory

Ahlswede, Rudolf

doi:10.1007/978-3-319-53139-7_6

Cited by 2 publications

(4 citation statements)

References 88 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The compatibility condition of ( 15) and ( 17) leads to the extended Lotka-Volterra lattice (12). This implies that the Lax pair of the finite extended Lotka-Volterra lattice reads…”

Section: If the Weight Functionmentioning

confidence: 99%

“…In this section, we adopt Moser's approach [14,22] and Bogoyavlensky's method [28] to introduce a set of rational functions to give an integrable linearization of the extended Lotka-Volterra lattice (12). Assume that all eigenvalues of (19) are simple.…”

Section: Linearization Of Extended Lotka-volterra Latticementioning

confidence: 99%

“…In fact, the famous Berlekamp-Massey algorithm in coding theory [9,10] may naturally follow from solving the problem of rational approximant of the formal Laurent series over a finite field. These works promote the development of other related subjects such as orthogonal polynomials [11,12], combinatorics [10], continued fractions [13] etc.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Decoding algorithm as a moment problem related to the extended Lotka–Volterra system

Pan¹,

Chang²,

Hu³

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The moment problem related to the extended Lotka-Volterra system (sometimes also called the hungry Lotka-Volterra system or the Narita-Itoh-Bogoyavlensky lattice) over finite fields is introduced. It turns out the moment problem could be used to design an algorithm for decoding multiple BCH-Goppa codes over the same finite field simultaneously. When multiple codes have the same error locations, the algorithm requires fewer known syndromes and has lower computational complexity than running the decoding algorithm in Nakamura (1996 Phys. Lett. A 223 75-81) multiple times.

show abstract

“…The compatibility condition of ( 15) and ( 17) leads to the extended Lotka-Volterra lattice (12). This implies that the Lax pair of the finite extended Lotka-Volterra lattice reads…”

Section: If the Weight Functionmentioning

confidence: 99%

Section: Linearization Of Extended Lotka-volterra Latticementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Decoding algorithm as a moment problem related to the extended Lotka–Volterra system

Pan¹,

Chang²,

Hu³

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Polynomials have been a staple tool in the world of approximation, with applications spanning physics (Simon, 1971;Vinck et al, 2012), random matrix theory (Deift, 2000), statistics (Walter, 1977;Diaconis et al, 2008) and information theory (Dehesa et al, 2001;Ahlswede, 2017). However, applications of orthogonal polynomials to machine learning are scarce to the best of our knowledge.…”

Section: Polynomials In Bayesian Problems and Machine Learningmentioning

confidence: 99%

Orthogonal Polynomials Quadrature Algorithm (OPQA): A Functional Analytical Approach to Bayesian Inference

Wong¹

2022

Preprint

View full text Add to dashboard Cite

In this paper, we present the new Orthogonal Polynomials-Quadrature Algorithm (OPQA), a parallelizable algorithm that estimates both the posterior and the evidence in a Bayesian analysis in one pass by means of a functional analytic approach. First, OPQA relates the evidence to an orthogonal projection onto a special basis of our construct. Second, it lays out a fast and accurate computational scheme to compute the transform coefficients.OPQA can be summarized as follows. First, we consider the L 2 space associated with a measure with exponential weights. Then we construct a multivariate orthogonal basis which is dense in this space, such density being guaranteed by the Riesz's Theorem. As we project the square root of the joint distribution onto this basis of our choice, the density of the basis allows us to invoke the Parseval Identity, which equates the evidence with the sum of squares of the transform coefficients of this orthogonal projection. To compute those transform coefficients, we propose a computational scheme using Gauss-Hermite quadrature in higher dimensions. Not only does this approach avoids the potential high variance problem associated with random sampling methods, it significantly reduces the complexity of the computation and enables one to speed up the computational speed by parallelization.This new algorithm does not make any assumption about the independence of the latent variable, nor do we assume any knowledge of the prior. It solves for both the evidence and the posterior in one pass. An outline of the theoretical proof of the supporting algorithm will be provided.

show abstract

Orthogonal Polynomials in Information Theory

Cited by 2 publications

References 88 publications

Decoding algorithm as a moment problem related to the extended Lotka–Volterra system

Decoding algorithm as a moment problem related to the extended Lotka–Volterra system

Orthogonal Polynomials Quadrature Algorithm (OPQA): A Functional Analytical Approach to Bayesian Inference

Contact Info

Product

Resources

About